Factorization by elementary matrices, null-homotopy and products of exponentials for invertible matrices over rings

Abstract

Let R be a commutative unital ring. A well-known factorization problem is whether any matrix in SLn(R) is a product of elementary matrices with entries in R. To solve the problem, we use two approaches based on the notion of the Bass stable rank and on construction of a null-homotopy. Special attention is given to the case, where R is a ring or Banach algebra of holomorphic functions. Also, we consider a related problem on representation of a matrix in GLn(R) as a product of exponentials.

Exponential factorizations of holomorphic maps

“…and w = 0 is a solution to (2). For the second case n(a − d) n(c), we find similarly a solution u = 1, v = 0 and w = − a−d c to (2).…”

“…Theorem 1 improves a result of Doubtsov and Kutzschebauch, who showed the same result with three instead of two factors in the conclusion, see [2,Proposition 3]. Stated differently, Theorem 1 says that every element of SL 2 (O(X)) can be written as a product of two exponentials, where O(X) denotes the ring of holomorphic functions on a given open Riemann surface X.…”

“…and w = 0 is a solution to (2). For the second case n(a − d) n(c), we find similarly a solution u = 1, v = 0 and w = − a−d c to (2). The remaining case is n(a − d) < min(n(b), n(c)), which implies n(a − d) < n(b + c) and hence − b+c a−d is holomorphic in a neighborhood of the origin and vanishes at the origin.…”

“…For a Stein space , a complex Lie group and its exponential map , we say that a holomorphic map is a product of exponentials if there are holomorphic maps such that It is easy to see that any map which is a product of exponentials (for some sufficiently large ) is null‐homotopic. In the case where is the special linear group , the converse follows from as explained in . However, it turns out to be a difficult problem to determine the minimal number of needed factors in dependence of the dimensions of and .…”

“…It is easy to see that any map f which is a product of exponentials (for some sufficiently large k) is null-homotopic. In the case where G is the special linear group SL n (C), the converse follows from [5] as explained in [2]. However, it turns out to be a difficult problem to determine the minimal number k of needed factors in dependence of the dimensions of X and SL n (C).…”